Under the basic semantics (another semantics proposed by Dung -
see [9,8] - which is not presented here and which
has only one extension).

Looking more closely, we can prove the following result (proof in Appendix A):

Property 14Under the stable semantics, the class of the uni-accepted arguments
coincides with the class of the cleanly-accepted arguments.

Then, using a result issued from [10,11] and reused
in [8] which shows that, when there is no odd cycle, all
the preferred extensions are stable27, we
apply Property 14 and we obtain the following
consequence:

Consequence 1Under the preferred semantics, when there is no odd cycle, the class
of the uni-accepted arguments coincides with the class of the
cleanly-accepted arguments.

Finally, the exploitation of the gradual interaction-based valuations (see
Section 3) allows us to define new levels of collective
acceptability.

Let be a gradual valuation and let be the associated
preordering (partial or complete) on . This preordering can be
used inside each acceptability level (for example, the level of the
exi-accepted arguments) in order to identify arguments which are
better accepted than others.

Example 9 (continuation)Two different
gradual valuations are applied on the same graph:

With the instance of the generic valuation proposed in [4]
(see Section 3.1), we obtain the following comparisons:

With the global valuation with tuples presented in Section 3.2, we
obtain the following comparisons:

So, all the arguments belonging to a cycle are incomparable with
, , , and, even between them, there are few
comparison results.

If we apply the preordering induced by a valuation without respecting
the acceptability levels defined in this section,
counter-intuitive situations may happen. In Example 9, we
obtain:

With the valuation of [4] and under the preferred
semantics, despite the fact that is uni-accepted and
is only-exi-accepted.

With the valuation with tuples and under the preferred
semantics, despite the fact that is
only-exi-accepted and is not-accepted.

These counter-intuitive situations illustrate the difference between
the acceptability definition and the valuation definitions (even if
both use the interaction between arguments, they do not use it in the
same way).